# Cesàro summability

Cesàro summability is a generalized convergence criterion for infinite series. We say that a series $\sum_{n=0}^{\infty}a_{n}$ is Cesàro summable if the Cesàro means of the partial sums converge to some limit $L$. To be more precise, letting

 $s_{N}=\sum_{n=0}^{N}a_{n}$

denote the $N^{\text{th}}$ partial sum, we say that $\sum_{n=0}^{\infty}a_{n}$ Cesàro converges to a limit $L$, if

 $\frac{1}{N+1}(s_{0}+\ldots+s_{N})\rightarrow L\quad\text{as}\quad N\rightarrow\infty.$

Cesàro summability is a generalization of the usual definition of the limit of an infinite series.

###### Proposition 1

Suppose that

 $\sum_{n=0}^{\infty}a_{n}=L,$

in the usual sense that $s_{N}\rightarrow L$ as $N\rightarrow\infty$. Then, the series in question Cesàro converges to the same limit.

The converse, however is false. The standard example of a divergent series, that is nonetheless Cesàro summable is

 $\sum_{n=0}^{\infty}(-1)^{n}.$

The sequence of partial sums $1,0,1,0,\ldots$ does not converge. The Cesàro means, namely

 $\frac{1}{1},\frac{1}{2},\frac{2}{3},\frac{2}{4},\frac{3}{5},\frac{3}{6},\ldots$

do converge, with $1/2$ as the limit. Hence the series in question is Cesàro summable.

There is also a relation between Cesàro summability and Abel summability11This and similar results are often called Abelian theorems..

###### Theorem 2 (Frobenius)

A series that is Cesàro summable is also Abel summable. To be more precise, suppose that

 $\frac{1}{N+1}(s_{0}+\ldots+s_{N})\rightarrow L\quad\text{as}\quad N\rightarrow\infty.$

Then,

 $f(r)=\sum_{n=0}^{\infty}a_{n}r^{n}\rightarrow L\quad\text{as}\quad r% \rightarrow 1^{-}$

as well.

Title Cesàro summability CesaroSummability 2013-03-22 13:07:01 2013-03-22 13:07:01 rmilson (146) rmilson (146) 6 rmilson (146) Definition msc 40G05 CesaroMean